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# Chapter 4: Introduction to Rotating Machines

Module by: NGUYEN Phuc. E-mail the author

Chapter 4: Introduction to Rotating Machines

This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy

• The objective of this chapter is to introduce and discuss some of the principles underlying the performance of electric machinery, both ac and dc machines.

§4.1 Elementary Concepts

• Voltages can be induced by time-varying magnetic fields. In rotating machines, voltages are generated in windings or groups of coils by rotating these windings mechanically through a magnetic field, by mechanically rotating a magnetic field past the winding, or by designing the magnetic circuit so that the reluctance varies with rotation of the rotor.
• The flux linking a specific coil is changed cyclically, and a time-varying voltage is generated.
• Electromagnetic energy conversion occurs when changes in the flux linkage result from mechanical motion.
• A set of such coils connected together is typically referred to as an armature winding, a winding or a set of windings carrying ac currents.
• In ac machines such as synchronous or induction machines, the armature winding is typically on the stator. (the stator winding)
• In dc machines, the armature winding is found on the rotor. (the rotor winding)
• Synchronous and dc machines typically include a second winding (or set of windings), referred to as the field winding, which carrys dc current and which are used to produce the main operating flux in the machine.
• In dc machines, the field winding is found on the stator.
• In synchronous machines, the field winding is found on the rotor.
• Permanent magnets can be used in the place of field windings.
• In most rotating machines, the stator and rotor are made of electrical steel, and the windings are installed in slots on these structures.The stator and rotor structures are typically built from thin laminations of electrical steel, insulated from each other, to reduce eddy-current losses.

§4.2 Introduction to AC And DC Machines

§4.2.1 AC Machines

• Traditional ac machines fall into one of two categories: synchronous and induction.
• In synchronous machines, rotor-winding currents are supplied directly from the stationary frame through a rotating contact.
• In induction machines, rotor currents are induced in the rotor windings by a combination of the time-variation of the stator currents and the motion of the rotor relative to the stator.
• Synchronous Machines

Fig. 4.1: a simplified salient-pole ac synchronous generator with two poles.

• The armature winding is on the stator, and the field winding is on the rotor.
• The field winding is excited by direct current conducted to it by means of stationary carbon brushes that contact rotating slip rings or collector rings.
• It is advantages to have the single, low-power field winding on the rotor while having the high-power, typically multiple-phase, armature winding on the stator.
• Armature winding (a,a) consists of a single coil of N turns.
• Conductors forming these coil sides are connected in series by end connections.
• The rotor is turned at a constant speed by a source of mechanical power connected to its shaft. Flux paths are shown schematically by dashed lines.

Figure 4.1 Schematic view of a simple, two-pole, single-phase synchronous generator.

Assume a sinusoidal distribution of magnetic flux in the air gap of the machine in Fig.4.1.

• The radial distribution of air-gap flux density B is shown in Fig. 4.2(a) as a function of the spatial angle θθ size 12{θ} {} around the rotor periphery.
• As the rotor rotates, the flux –linkages of the armature winding change with time and the resulting coil voltage will be sinusoidal in time as shown in Fig 4.2(b). The frequency in cycles per second (Hz) is the same as the speed of the rotor in revolutions in second (rps).
• A two-pole synchronous machine must revolve at 3600 rpm to produce a 60Hz voltage.
• Note the terms “rpm” and “rps”.

Figure 4.2 (a) Space distribution of flux density and (b) corresponding waveform of

the generated voltage for the single-phase generator of Fig. 4.1.

A great many synchronous machines have more than two poles. Fig 4.3 shows in schematic form a four-pole single-phase generator.

• The field coils are connected so that the poles are of alternate polarity.
• The armature winding consists of two coils (a1,a1)(a1,a1) size 12{ $$a rSub { size 8{1} } , - a rSub { size 8{1} }$$ } {}and (a2,a2)(a2,a2) size 12{ $$a rSub { size 8{2} } , - a rSub { size 8{2} }$$ } {}connected in series by their end connections.
• There are two complete wavelengths, or cycles, in the flux distribution around the periphery, as shown in Fig. 4.4.
• The generated voltage goes through two complete cycles per revolution of the rotor.
• The frequency in Hz is thus twice the speed in rps.

Figure 4.3 Schematic view of a simple, four-pole, single-phase synchronous generator.

Figure 4.4 Space distribution of the air-gap flux density in an idealized,

four-pole synchronous generator.

When a machine has more than two poles, it is convenient to concentrate on a single pair of poles and to express angles in electrical degrees or electrical radians rather than in physical units.

• One pair of poles equals 360 electrical degrees or 2 ππ size 12{π} {} electrical radians.
• Since there are poles/2 wavelengths, or cycles, in one revolution, it follows that

θae=(poles2)θaθae=(poles2)θa size 12{θ rSub { size 8{ ital "ae"} } = $${ { ital "poles"} over {2} }$$ θ rSub { size 8{a} } } {} (4.1)

Where θaeθae size 12{θ rSub { size 8{ ital "ae"} } } {}is the angle in electrical units and θaθa size 12{θ rSub { size 8{a} } } {} is the spatial angle.

• The coil voltage of a multipole machine passes through a complete cycle every time a pair of poles sweeps by, or (poles/2) times each revolution. The electrical frequency fefe size 12{f rSub { size 8{e} } } {} of the voltage generated is therefore

fe=(poles2)n60 Hzfe=(poles2)n60 Hz size 12{f rSub { size 8{e} } = $${ { ital "poles"} over {2} }$$ { {n} over {"60"} } " Hz"} {} (4.2)

where n is the mechanical speed in rpm.Note that ωe=(poles/2)ωmωe=(poles/2)ωm size 12{ω rSub { size 8{e} } = $$ital "poles"/2$$ ω rSub { size 8{m} } } {}

• The rotors shown in Figs.4.1 and 4.3 have salient, or projecting, poles with concentrated windings. Fig.4.5 shows diagrammatically a nonsalient-pole, or cylindrical, rotor.
• The field winding is a two-pole distributed winding; the coil sides are distributed in multiple slots around the rotor periphery and arranged to produce an approximately sinusoidal distribution of radial air-gap flux.
• Most power systems in the world operate at frequencies of either 50 or 60 Hz.
• A salient-pole construction is characteristic of hydroelectric generators because hydraulic turbines operate at relatively low speeds, and hence a relatively large number of poles is required to produce the desired frequency.
• Steam turbines and gas turbines operate best at relatively high speeds, and turbine- driven alternators or turbine generators are commonly two- or four-pole cylindrical- rotor machines.

Figure 4.5 Elementary two-pole cylindrical-rotor field winding.

Most of the world’s power systems are three-phase systems. With very few exceptions, synchronous generators are three-phase machines.

• A simplified schematic view of a three-phase, two-pole machine with one coil per phase is shown in Fig. 4.6 (a)
• Fig. 4.6(b) depicts a simplified three-phase, four-pole machine. Note that a minimum of two sets of coils must be used. In an elementary multipole machine, the minimum number of coils sets is given by one half the number of poles.
• Note that coils (a,a) and (a',a')(a',a') size 12{ $${ {a}} sup { ' }, - { {a}} sup { ' }$$ } {} can be connected in series or in parallel. Then the coils of the three phases may then be either Y- or ΔΔ size 12{Δ} {}-connected. See Fig. 4.6(c).

Figure 4.6 Schematic views of three-phase generators: (a) two-pole, (b) four-pole, and

(c) Y connection of the windings.

• The electromechanical torque is the mechanism through which a synchronous generator converts mechanical to electric energy.
• When a synchronous generator supplies electric power to a load, the armature current creates a magnetic flux wave in the air gap that rotates at synchronous speed.
• This flux reacts with the flux created by the field current, and an electromechanical torque results from the tendency of these two magnetic fields to align.
• In a generator this torque opposes rotation, and mechanical torque must be applied from the prime mover to sustain rotation.
• The counterpart of the synchronous generator is the synchronous motor.
• Ac current supplied to the armature winding on the stator, and dc excitation is supplied to the field winding on the rotor. The magnetic field produced by the armature currents rotates at synchronous speed.
• To produce a steady electromechanical torque, the magnetic fields of the stator and rotor must be constant in amplitude and stationary with respect to each other.
• In a motor the electromechanical torque is in the direction of rotation and balances the opposing torque required to drive the mechanical load.
• In both generators and motors, an electromechanical torque and a rotational voltage are produced which are the essential phenomena for electromechanical energy conversion.
• Note that the flux produced by currents in the armature of a synchronous motor rotates ahead of that produced by the field, thus pulling on the field (and hence on the rotor) and doing work. This is the opposite of the situation in a synchronous generator, where the field does work as its flux pulls on that of the armature, which is lagging behind.
• Induction Machines
• Alternating currents are applied directly to the stator windings. Rotors currents are then produced by induction, i.e., transformer action.
• Alternating currents flow in the rotor windings of an induction machine, in contrast to a synchronous machine in which a field winding on the rotor is excited with dc current.
• The induction machine may be regarded as a generalized transformer in which electric power is transformed between rotor and stator together with a change of frequency and a flow of mechanical power.
• The induction motor is the most common of all motors.
• The induction machine is seldom used as a generator.
• In recent years it has been found to be well suited for wind-power applications.
• It may also be used as a frequency changer.
• In the induction motor, the stator windings are essentially the same as those of a synchronous machine.The rotor windings are electrically short-circuited.
• The rotor windings frequently have no external connections.
• Currents are induced by transformer action from the stator winding.
• Squirrel-cage induction motor: relatively expensive and highly reliable.
• The armature flux in the induction motor leads that of the rotor and produces an electromechanical torque.
• The rotor does not rotate synchronously.
• It is the slipping of the rotor with respect to the synchronous armature flux that gives rise to the induced rotor currents and hence the torque.
• Induction motors operate at speeds less than the synchronous mechanical speed.
• A typical speed-torque characteristic for an induction motor is shown in Fig.4.7.

Figure 4.7 Typical induction-motor speed-torque characteristic.

§4.2.2 DC Machines

• DC Machines
• There are two sets of windings in a dc machine.
• The armature winding is on the rotor with current conducted from it by means of carbon brushes.
• The field winding is on the stator and is excited by direct current.
• An elementary two-pole dc generator is shown in Fig. 4.8.
• Armature winding: (a,a) , pitch  180o180o size 12{"180" rSup { size 8{o} } } {}
• The rotor is normally turned at a constant speed by a source of mechanical power connected the shaft.

Figure 4.8 Elementary dc machine with commutator.

• The air-gap flux distribution usually approximates a flat-topped wave, rather than the sine wave found in ac machines, and is shown in Fig. 4.9(a).
• Rotation of the coil generates a coil voltage which is a time function having the same waveform as the spatial flux-density distribution.
• The voltage induced in an individual armature coil is an alternating voltage and rectification is produced mechanically by means of a commutator. Stationary carbon brushes held against the commutator surface connect the winding to the external armature terminal.
• The need for commutation is the reason why the armature windings are placed on the rotor.
• The commutator provides full-wave rectification, and the voltage waveform between brushes is shown in Fig. 4.9(b).

Figure 4.9 (a) Space distribution of air-gap flux density in an elementary dc machine;

(b) waveform of voltage between brushes.

It is the interaction of the two flux distributions created by the direct currents in the field and the armature windings that creates an electromechanical torque.

• If the machine is acting as a generator, the torque opposes rotation.
• If the machine is acting as a motor, the torque acts in the direction of the rotation.

§4.3 MMF of Distributed Windings

• Most armatures have distributed windings, i.e. windings which are spread over a number of slots around the air-gap periphery.
• The individual coils are interconnected so that the result is a magnetic field having the same number of poles as the field winding.
• Consider Fig. 4.10(a).
• Full-pitch coil: a coil which spans 180 electrical degrees.
• In Fig. 4.10(b), the air gap and winding are in developed form (laid out flat) and the air-gap mmf distribution is shown by the steplike distribution of amplitude

Figure 4.10 (a) Schematic view of flux produced by a concentrated, full-pitch winding in a machine with a uniform air gap. (b) The air-gap mmf produced by current in this winding.

§4.3.1 AC Machines

• It is appropriate to focus our attention on the space-fundamental sinusoidal component of the air-gap mmf.
• In the design of ac machines, serious efforts are made to distribute the coils making up the windings so as to minimize the higher-order harmonic components.
• The rectangular air-gap mmf wave of the concentrated two-pole, full-pitch coil of Fig.4.10(b) can be resolved to a Fourier series comprising a fundamental component and a series of odd harmonics.
• The fundamental component FaglFagl size 12{F rSub { size 8{ ital "agl"} } } {}and its amplitude (Fagl)peak(Fagl)peak size 12{ $$F rSub { size 8{ ital "agl"} }$$ rSub { size 8{ ital "peak"} } } {}are

Fagl=4π(Ni2)cosθaFagl=4π(Ni2)cosθa size 12{F rSub { size 8{ ital "agl"} } = { {4} over {π} } $${ { ital "Ni"} over {2} }$$ "cos"θ rSub { size 8{a} } } {} (4.3)

(Fagl)peak=4π(Ni2)(Fagl)peak=4π(Ni2) size 12{ $$F rSub { size 8{ ital "agl"} }$$ rSub { size 8{ ital "peak"} } = { {4} over {π} } $${ { ital "Ni"} over {2} }$$ } {} (4.4)

Consider a distributed winding, consisting of coils distributed in several slots.

• Fig. 4.11(a) shows phase a of the armature winding of a simplified two-pole, three-phase ac machine and phases b and c occupy the empty slots.
• The windings of the three phases are identical and are located with their magnetic axes 120 degrees apart.The winding is arranged in two layers, each full-pitch coil of NcNc size 12{N rSub { size 8{c} } } {} turns having one side in the top of a slot and the other coil side in the bottom of a slot a pole pitch away.
• Fig. 4.11(b) shows that the mmf wave is a series of steps each of height 2Ncia2Ncia size 12{2N rSub { size 8{c} } i rSub { size 8{a} } } {}. It can be seen that the distributed winding produces a closer approximation to a sinusoidal mmf wave than the concentrated coil of Fig.4.10 does.

Figure 4.11 The mmf of one phase of a distributed two-pole,

three-phase winding with full-pitch coils.

• The modified form of (4.3) for a distributed multipole winding is

Fagl=4π(kwNphpoles)iacos(poles2θa)Fagl=4π(kwNphpoles)iacos(poles2θa) size 12{F rSub { size 8{ ital "agl"} } = { {4} over {π} } $${ {k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } } over { ital "poles"} }$$ i rSub { size 8{a} } "cos" $${ { ital "poles"} over {2} } θ rSub { size 8{a} }$$ } {} (4.5)

NphNph size 12{N rSub { size 8{ ital "ph"} } } {}: number of series turns per phase,

kwkw size 12{k rSub { size 8{w} } } {}: winding factor, a reduction factor taking into account the distribution of the winding, typically in the range of 0.85 to 0.95, kw=kbkp(orkdkp)kw=kbkp(orkdkp) size 12{k rSub { size 8{w} } =k rSub { size 8{b} } k rSub { size 8{p} } $$ital "or" k rSub { size 8{d} } k rSub { size 8{p} }$$ } {}.

• The peak amplitude of this mmf wave is

(Fagl)peak=4π(kwNphpoles)ia(Fagl)peak=4π(kwNphpoles)ia size 12{ $$F rSub { size 8{ ital "agl"} }$$ rSub { size 8{ ital "peak"} } = { {4} over {π} } $${ {k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } } over { ital "poles"} }$$ i rSub { size 8{a} } } {} (4.6)

• Eq. (4.5) describes the space-fundamental component of the mmf wave produced by current in phase a of a distributed winding.
• If ia=Imcosωtia=Imcosωt size 12{i rSub { size 8{a} } =I rSub { size 8{m} } "cos"ωt} {} the result will be an mmf wave which is stationary in space and varies sinusoidally both with respect to θaθa size 12{θ rSub { size 8{a} } } {} and in time.
• The application of three-phase currents will produce a rotating mmf wave.
• Rotor windings are often distributed in slots to reduce the effects of space harmonics.
• Fig. 4.12(a) shows the rotor of a typical two-pole round-rotor generator.
• As shown in Fig. 4.12(b), there are fewer turns in the slots nearest the pole face.
• The fundamental air-gap mmf wave of a multipole rotor winding is

Fagl=4π(krNrpoles)Ircos(poles2θr)Fagl=4π(krNrpoles)Ircos(poles2θr) size 12{F rSub { size 8{ ital "agl"} } = { {4} over {π} } $${ {k rSub { size 8{r} } N rSub { size 8{r} } } over { ital "poles"} }$$ I rSub { size 8{r} } "cos" $${ { ital "poles"} over {2} } θ rSub { size 8{r} }$$ } {} (4.7)

(Fagl)peak=4π(krNrpoles)Ir(Fagl)peak=4π(krNrpoles)Ir size 12{ $$F rSub { size 8{ ital "agl"} }$$ rSub { size 8{ ital "peak"} } = { {4} over {π} } $${ {k rSub { size 8{r} } N rSub { size 8{r} } } over { ital "poles"} }$$ I rSub { size 8{r} } } {} (4.8)

Figure 4.12 The air-gap mmf of a distributed winding on the rotor of a round-rotor generator.

§4.3.2 DC Machines

• Because of the restrictions imposed on the winding arrangement by the commutator, the mmf wave of a dc machine armature approximates a sawtooth waveform more nearly than the sine wave of ac machines.
• Fig. 4.13 shows diagrammatically in cross section the armature of a two-pole dc machine.
• The armature coil connections are such that the armature winding produces a magnetic field whose axis is vertical and thus is perpendicular to the axis of the field winding.
• As the armature rotates, the magnetic field of the armature remains vertical due to commutator action and a continuous unidirectional torque results.
• The mmf wave is illustrated and analyzed in Fig. 4.14.

Figure 4.13 Cross section of a two-pole dc machine.

Figure 4.14 (a) Developed sketch of the dc machine of Fig. 4.22; (b) mmf wave; (c) equivalent sawtooth mmf wave, its fundamental component, and equivalent rectangular current sheet.

DC machines often have a magnetic structure with more than two poles.

• Fig. 4.15(a) shows schematically a four-pole dc machine.
• The machine is shown in laid-out form in Fig. 4.15(b).

Figure 4.15 (a) Cross section of a four-pole dc machine; (b) development of current sheet and mmf wave.

• The peak value of the sawtooth armature mmf wave can be written as

(Fag)peak=(Ca2m.poles)ia A.turn/ pole(Fag)peak=(Ca2m.poles)ia A.turn/ pole size 12{ $$F rSub { size 8{ ital "ag"} }$$ rSub { size 8{ ital "peak"} } = $${ {C rSub { size 8{a} } } over {2m "." ital "poles"} }$$ i rSub { size 8{a} } " A" "." "turn/ pole"} {} (4.9)

Ca = total number of conductors in armature winding

m = number of parallel paths through armature winding

ia = armature current, A

(Fag)peak=(Napoles)ia,Na=Ca/(2m):no. of series armature turns(Fag)peak=(Napoles)ia,Na=Ca/(2m):no. of series armature turns size 12{ $$F rSub { size 8{ ital "ag"} }$$ rSub { size 8{ ital "peak"} } = $${ {N rSub { size 8{a} } } over { ital "poles"} }$$ i rSub { size 8{a} } ," "N rSub { size 8{a} } =C rSub { size 8{a} } / $$2m$$ :"no" "." " of series armature turns"} {} (4.10)

(Fag)peak=8π2(Napoles)ia(Fag)peak=8π2(Napoles)ia size 12{ $$F rSub { size 8{ ital "ag"} }$$ rSub { size 8{ ital "peak"} } = { {8} over {π rSup { size 8{2} } } } $${ {N rSub { size 8{a} } } over { ital "poles"} }$$ i rSub { size 8{a} } } {} (4.11)

§4.4 Magnetic Fields In Rotating Machinery

• The behavior of electric machinery is determined by the magnetic fields created by currents in the various windings of the machine.
• The investigations of both ac and dc machines are based on the assumption of sinusoidal spatial distribution of mmf.
• Results from examining a two-pole machine can immediately be extrapolated to a multipole machine.

§4.4.1 Magnetic with Uniform Air Gaps

• Consider machines with uniform air gaps.
• Fig. 4.16(a) shows a single full-pitch, N-turn coil in a high-permeability magnetic structure μμ size 12{μ rightarrow infinity } {} , with a concentric, cylindrical rotor.
• In Fig. 4.16(b) the air-gap mmf FagFag size 12{F rSub { size 8{ ital "ag"} } } {}is plotted versus angle θaθa size 12{θ rSub { size 8{a} } } {}.
• Fig. 4.16(c) demonstrates the air-gap constant radial magnetic field HagHag size 12{H rSub { size 8{ ital "ag"} } } {}.

Hag=FaggHag=Fagg size 12{H rSub { size 8{ ital "ag"} } = { {F rSub { size 8{ ital "ag"} } } over {g} } } {} (4.12)

(Hagl)=Faglg=4π(Ni2g)cosθa(Hagl)=Faglg=4π(Ni2g)cosθa size 12{ $$H rSub { size 8{ ital "agl"} }$$ = { {F rSub { size 8{ ital "agl"} } } over {g} } = { {4} over {π} } $${ { ital "Ni"} over {2g} }$$ "cos"θ rSub { size 8{a} } } {} (4.13)

(Hagl)peak=4π(Ni2g)(Hagl)peak=4π(Ni2g) size 12{ $$H rSub { size 8{ ital "agl"} }$$ rSub { size 8{ ital "peak"} } = { {4} over {π} } $${ { ital "Ni"} over {2g} }$$ } {} (4.14)

• For a distributed winding, the air-gap magnetic field intensity is

Hagl=4π(kwNphg.poles)iacos(poles2θa)Hagl=4π(kwNphg.poles)iacos(poles2θa) size 12{H rSub { size 8{ ital "agl"} } = { {4} over {π} } $${ {k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } } over {g "." ital "poles"} }$$ i rSub { size 8{a} } "cos" $${ { ital "poles"} over {2} } θ rSub { size 8{a} }$$ } {} (4.15)

Figure 4.16 The air-gap mmf and radial component of HagHag size 12{H rSub { size 8{ ital "ag"} } } {} for a concentrated full-pitch winding.

§4.4.2 Machines with Nonuniform Air Gaps

• The air-gap magnetic-field distribution of machines with nonuniform air gaps is more complex than that of uniform-air-gap machines.
• Fig. 4.17(a) shows the structure of a typical dc machine and Fig. 4.17 (b) shows the structure of a typical salient-pole synchronous machine.

Figure 4.17 Structure of typical salient-pole machines:

(a) dc machine and (b) salient-pole synchronous machine.

• Detailed analysis of the magnetic field distributions requires complete solutions of the field problem.
• Fig. 4.18 shows the magnetic field distribution in a salient-pole dc generator (obtained by finite-element solution).

Figure 4.18 Finite-element solution of the magnetic field distribution in a salient-pole dc generator. Field coils excited; no current in armature coils. (General Electric Company.)

§4.5 Rotating MMF Waves in AC Machines

• To understand the theory and operation of polyphase ac machines, it is necessary to study the nature of the mmf wave produced by a polyphase winding.

§4.5.1 MMF Wave of a Single-Phase Winding

• Fig. 4.19(a) shows the space-fundamental mmf distribution of a single-phase winding.
• Note that from Eq. (4.5), FaglFagl size 12{F rSub { size 8{ ital "agl"} } } {} is

Fagl=4π(kwNphpoles)iacos(poles2θa)Fagl=4π(kwNphpoles)iacos(poles2θa) size 12{F rSub { size 8{ ital "agl"} } = { {4} over {π} } $${ {k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } } over { ital "poles"} }$$ i rSub { size 8{a} } "cos" $${ { ital "poles"} over {2} } θ rSub { size 8{a} }$$ } {} (4.16)

When the winding is exicted by a current

ia=Iacosωetia=Iacosωet size 12{i rSub { size 8{a} } =I"" lSub { size 8{a} } "cos"ω rSub { size 8{e} } t} {} (4.17)

the mmf distribution is given by

Fagl=Fmaxcos(poles2θa)cosωet=Fmaxcos(θae)cosωetFagl=Fmaxcos(poles2θa)cosωet=Fmaxcos(θae)cosωetalignl { stack { size 12{F rSub { size 8{ ital "agl"} } =F rSub { size 8{"max"} } "cos" $${ { ital "poles"} over {2} } θ rSub { size 8{a} }$$ "cos"ω rSub { size 8{e} } t} {} # " "=F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} }$$ "cos"ω rSub { size 8{e} } t {} } } {} (4.18)

Fmax=4π(kwNphpoles)IaFmax=4π(kwNphpoles)Ia size 12{F rSub { size 8{"max"} } = { {4} over {π} } $${ {k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } } over { ital "poles"} }$$ I rSub { size 8{a} } } {} (4.19)

• This mmf distribution remains fixed in space with an amplitude that varies sinusoidally in time at frequency ωcωc size 12{ω rSub { size 8{c} } } {} , as shown in Fig. 4.19(a).
• The air-gap mmf of a single-phase winding exicted by a source of ac current can be resolved into rotating traveling waves.
• By the identity cosαcosβ=12cos(αβ)+cos(α+β)cosαcosβ=12cos(αβ)+cos(α+β) size 12{"cos"α"cos"β= { {1} over {2} } "cos" $$α - β$$ +"cos" $$α+β$$ } {}

Fagl=Fmax[12cos(θaeωet)+12cos(θae+ωet)]Fagl=Fmax[12cos(θaeωet)+12cos(θae+ωet)] size 12{F rSub { size 8{ ital "agl"} } =F rSub { size 8{"max"} } ${ {1} over {2} } "cos" $$θ rSub { size 8{ ital "ae"} } - ω rSub { size 8{e} } t$$ + { {1} over {2} } "cos" $$θ rSub { size 8{ ital "ae"} } +ω rSub { size 8{e} } t$$$ } {} (4.20)

Fagl+=Fmaxcos(θaeωet)Fagl+=Fmaxcos(θaeωet) size 12{F rSub { size 8{ ital "agl"} } rSup { size 8{+{}} } =F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } - ω rSub { size 8{e} } t$$ } {} (4.21)

Fagl=12Fmaxcos(θae+ωet)Fagl=12Fmaxcos(θae+ωet) size 12{F rSub { size 8{ ital "agl"} } rSup { size 8{ - {}} } = { {1} over {2} } F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } +ω rSub { size 8{e} } t$$ } {} (4.22)

• FaglFagl size 12{F rSub { size 8{ ital "agl"} } } {}travels in the +θa+θa size 12{+θ rSub { size 8{a} } } {}direction and FaglFagl size 12{F rSub { size 8{ ital "agl"} } } {}travels in the θaθa size 12{ - θ rSub { size 8{a} } } {} direction.
• This decomposition is shown graphically in Fig. 4.19(b) and in a phasor representation in Fig. 4.19(c).

Figure 4.19 Single-phase-winding space-fundamental air-gap mmf: (a) mmf distribution of a

single-phase winding at various times; (b) total mmf FaglFagl size 12{F rSub { size 8{ ital "agl"} } } {}decomposed into two traveling waves F and F+F+ size 12{F rSup { size 8{+{}} } } {} ; (c) phasor decomposition of F.

§4.5.2 MMF Wave of a Polyphase Winding

• We are to study the mmf distribution of three-phase windings such as those found on the stator of three-phase induction and synchronous machines.

In a three-phase machine, the windings of the individual phases are displaced from each other by 120 electrical degrees in space around the air-gap circumference as shown in Fig.4.20 in which the concentrated full-pitch coils may be considered to represent distributed windings.

• Under balanced three-phase conditions, the excitation currents (Fig. 4.20) are

ia=Imcosωetia=Imcosωet size 12{i rSub { size 8{a} } =I rSub { size 8{m} } "cos"ω rSub { size 8{e} } t} {} (4.23)

ib=Imcos(ωet120o)ib=Imcos(ωet120o) size 12{i rSub { size 8{b} } =I rSub { size 8{m} } "cos" $$ω rSub { size 8{e} } t - "120" rSup { size 8{o} }$$ } {} (4.24)

ic=Imcos(ωet+120o)ic=Imcos(ωet+120o) size 12{i rSub { size 8{c} } =I rSub { size 8{m} } "cos" $$ω rSub { size 8{e} } t+"120" rSup { size 8{o} }$$ } {} (4.25)

Figure 4.20 Simplified two-pole three-phase stator winding.

Figure 4.21 Instantaneous phase currents under balanced three-phase conditions.

• The mmf of phase a has been shown to be

Fa1=Fa1++Fa1Fa1=Fa1++Fa1 size 12{F rSub { size 8{a1} } =F rSub { size 8{a1} } rSup { size 8{+{}} } +F rSub { size 8{a1} } rSup { size 8{ - {}} } } {} (4.26)

Fa1+=12Fmaxcos(θaeωet)Fa1+=12Fmaxcos(θaeωet) size 12{F rSub { size 8{a1} } rSup { size 8{+{}} } = { {1} over {2} } F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } - ω rSub { size 8{e} } t$$ } {} (4.27)

Fa1=12Fmaxcos(θae+ωet)Fa1=12Fmaxcos(θae+ωet) size 12{F rSub { size 8{a1} } rSup { size 8{ - {}} } = { {1} over {2} } F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } +ω rSub { size 8{e} } t$$ } {} (4.28)

Fmax=4π(kwNphpoles)ImFmax=4π(kwNphpoles)Im size 12{F rSub { size 8{"max"} } = { {4} over {π} } $${ {k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } } over { ital "poles"} }$$ I rSub { size 8{m} } } {} (4.29)

• Similarly, for phases b and c

Fb1=Fb1++Fb1Fb1=Fb1++Fb1 size 12{F rSub { size 8{b1} } =F rSub { size 8{b1} } rSup { size 8{+{}} } +F rSub { size 8{b1} } rSup { size 8{ - {}} } } {} (4.30)

Fb1+=12Fmaxcos(θaeωet)Fb1+=12Fmaxcos(θaeωet) size 12{F rSub { size 8{b1} } rSup { size 8{+{}} } = { {1} over {2} } F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } - ω rSub { size 8{e} } t$$ } {} (4.31)

Fb1=12Fmaxcos(θae+ωet+120o)Fb1=12Fmaxcos(θae+ωet+120o) size 12{F rSub { size 8{b1} } rSup { size 8{ - {}} } = { {1} over {2} } F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } +ω rSub { size 8{e} } t+"120" rSup { size 8{o} }$$ } {}(4.32)

Fc1=Fc1++Fc1Fc1=Fc1++Fc1 size 12{F rSub { size 8{c1} } =F rSub { size 8{c1} } rSup { size 8{+{}} } +F rSub { size 8{c1} } rSup { size 8{ - {}} } } {} (4.33)

Fc1+=12Fmaxcos(θaeωet)Fc1+=12Fmaxcos(θaeωet) size 12{F rSub { size 8{c1} } rSup { size 8{+{}} } = { {1} over {2} } F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } - ω rSub { size 8{e} } t$$ } {} (4.34)

Fc1=Fmaxcos(θae+ωet120o)Fc1=Fmaxcos(θae+ωet120o) size 12{F rSub { size 8{c1} } rSup { size 8{ - {}} } =F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } +ω rSub { size 8{e} } t - "120" rSup { size 8{o} }$$ } {}(4.35)

• The total mmf is the sum

F(θae,t)=Fa1+Fb1+Fc1F(θae,t)=Fa1+Fb1+Fc1 size 12{F $$θ rSub { size 8{ ital "ae"} } ,t$$ =F rSub { size 8{a1} } +F rSub { size 8{b1} } +F rSub { size 8{c1} } } {}(4.36)

It can be performed in terms of the positive- and negative- traveling waves.

F ( θ ae , t ) = F a1 + F b1 + F c1 = 1 2 F max [ cos ( θ ae + ω e t ) + cos ( θ ae + ω e t 120 o ) + cos ( θ ae + ω e t + 120 o ) ] = 0 F ( θ ae , t ) = F a1 + F b1 + F c1 = 1 2 F max [ cos ( θ ae + ω e t ) + cos ( θ ae + ω e t 120 o ) + cos ( θ ae + ω e t + 120 o ) ] = 0 alignl { stack { size 12{F rSup { size 8{ - {}} } $$θ rSub { size 8{ ital "ae"} } ,t$$ =F rSub { size 8{a1} } rSup { size 8{ - {}} } +F rSub { size 8{b1} } rSup { size 8{ - {}} } +F rSub { size 8{c1} } rSup { size 8{ - {}} } } {} # " "= { {1} over {2} } F rSub { size 8{"max"} } $"cos" $$θ rSub { size 8{ ital "ae"} } +ω rSub { size 8{e} } t$$ +"cos" $$θ rSub { size 8{ ital "ae"} } +ω rSub { size 8{e} } t - "120" rSup { size 8{o} }$$ +"cos" $$θ rSub { size 8{ ital "ae"} } +ω rSub { size 8{e} } t+"120" rSup { size 8{o} }$$$ {} # " "=0 {} } } {}

F+(θae,t)=Fa1++Fb1++Fc1+=32Fmaxcos(θaeωet)F+(θae,t)=Fa1++Fb1++Fc1+=32Fmaxcos(θaeωet)alignl { stack { size 12{F rSup { size 8{+{}} } $$θ rSub { size 8{ ital "ae"} } ,t$$ =F rSub { size 8{a1} } rSup { size 8{+{}} } +F rSub { size 8{b1} } rSup { size 8{+{}} } +F rSub { size 8{c1} } rSup { size 8{+{}} } } {} # " "= { {3} over {2} } F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } - ω rSub { size 8{e} } t$$ {} } } {} (4.37)

• The result of displacing the three windings by 120o120o size 12{"120" rSup { size 8{o} } } {} in space phase and displacing the winding currents by 120o120o size 12{"120" rSup { size 8{o} } } {} in time phase is a single positive-traveling mmf wave

F(θae,t)=32Fmaxcos(θaeωet)32Fmaxcos((poles2)θaωet)F(θae,t)=32Fmaxcos(θaeωet)32Fmaxcos((poles2)θaωet)alignl { stack { size 12{F $$θ rSub { size 8{ ital "ae"} } ,t$$ = { {3} over {2} } F rSub { size 8{"max"} } "cos" $$θ rSub { size 8{ ital "ae"} } - ω rSub { size 8{e} } t$$ } {} # = { {3} over {2} } F rSub { size 8{"max"} } "cos" $$\( { { ital "poles"} over {2} }$$ θ rSub { size 8{a} } - ω rSub { size 8{e} } t \) {} } } {} (4.38)

• Under balanced three-phase conditions, the three-phase winding produces an air-gap mmf wave which rotates at synchronous angular velocity ωsωs size 12{ω rSub { size 8{s} } } {} (rad/sec)

ωs=(2poles)ωeωs=(2poles)ωe size 12{ω rSub { size 8{s} } = $${ {2} over { ital "poles"} }$$ ω rSub { size 8{e} } } {} (4.39)

ωcωc size 12{ω rSub { size 8{c} } } {}: angular velocity of the applied electrical excitation (rad/sec)

• : synchronous speed

fe=ωe/()fe=ωe/() size 12{f rSub { size 8{e} } =ω rSub { size 8{e} } / $$2π$$ } {}: applied electrical frequency

ns=(120poles)fer/minns=(120poles)fer/min size 12{n rSub { size 8{s} } = $${ {"120"} over { ital "poles"} }$$ f rSub { size 8{e} } " "r"/min"} {} (4.40)

• A polyphase winding exicted by balanced polyphase currents produces a rotating mmf wave.
• It is the interaction of this magnetic flux wave with that of the rotor which produces torque.
• Constant torque is produced when rotor-produced magnetic flux rotates in synchronism with that of the stator.

§4.5.3 Graphical Analysis of Polyphase MMF

• For balanced three-phase currents, the production of a rotating mmf can also be shown graphically.
• Refer to Fig. 4.22.
• As time passes, the resultant mmf wave retains its sinusoidal form and amplitude but rotates progressively around the air gap.
• The net result is an mmf wave of constant amplitude rotating at uniform angular velocity.

Figure 4.22 The production of a rotating magnetic field by means of three-phase currents.

§4.6 Generated Voltage

§4.6.1 AC Machines

Figure 4.23 Cross-sectional view of an elementary three-phase ac machine.

Bpeak=oπg(kfNfpoles)IfBpeak=oπg(kfNfpoles)If size 12{B rSub { size 8{ ital "peak"} } = { {4μ rSub { size 8{o} } } over {πg} } $${ {k rSub { size 8{f} } N rSub { size 8{f} } } over { ital "poles"} }$$ I rSub { size 8{f} } } {} (4.41)

B=Bpeakcos(poles2θr)B=Bpeakcos(poles2θr) size 12{B=B rSub { size 8{ ital "peak"} } "cos" $${ { ital "poles"} over {2} } θ rSub { size 8{r} }$$ } {} (4.42)

Φ=lπ/poles+π/polesBpeakcos(poles2θr)rdθr=(2poles)2BpeaklrΦ=lπ/poles+π/polesBpeakcos(poles2θr)rdθr=(2poles)2Bpeaklralignl { stack { size 12{Φ=l Int rSub { size 8{ - π/ ital "poles"} } rSup { size 8{+π/ ital "poles"} } {B rSub { size 8{ ital "peak"} } } "cos" $${ { ital "poles"} over {2} } θ rSub { size 8{r} }$$ ital "rd"θ rSub { size 8{r} } } {} # " "= $${ {2} over { ital "poles"} }$$ 2B rSub { size 8{ ital "peak"} } ital "lr" {} } } {} (4.43)

λa=kwNphΦpcos((poles2)ωmt)=kwNphΦpcosωmetλa=kwNphΦpcos((poles2)ωmt)=kwNphΦpcosωmetalignl { stack { size 12{λ rSub { size 8{a} } =k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } "cos" $$\( { { ital "poles"} over {2} }$$ ω rSub { size 8{m} } t \) } {} # " "=k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } "cos"ω rSub { size 8{ ital "me"} } t {} } } {} (4.44)

ωme=(poles2)ωmωme=(poles2)ωm size 12{ω rSub { size 8{ ital "me"} } = $${ { ital "poles"} over {2} }$$ ω rSub { size 8{m} } } {} (4.45)

ea=adt=kwNphpdtcosωmetωmekwNphΦpsinωmetea=adt=kwNphpdtcosωmetωmekwNphΦpsinωmet size 12{e rSub { size 8{a} } = { {dλ rSub { size 8{a} } } over { ital "dt"} } =k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } { {dΦ rSub { size 8{p} } } over { ital "dt"} } "cos"ω rSub { size 8{ ital "me"} } t - ω rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } "sin"ω rSub { size 8{ ital "me"} } t} {} (4.46)

ea=ωmekwNphΦpsinωmetea=ωmekwNphΦpsinωmet size 12{e rSub { size 8{a} } = - ω rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } "sin"ω rSub { size 8{ ital "me"} } t} {} (4.47)

Emax=ωmekwNphΦp=2πfmekwNphΦpEmax=ωmekwNphΦp=2πfmekwNphΦp size 12{E rSub { size 8{"max"} } =ω rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } =2πf rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } } {} (4.48)

Erms=2fmekwNphΦp=2fmekwNphΦpErms=2fmekwNphΦp=2fmekwNphΦp size 12{E rSub { size 8{ ital "rms"} } = { {2π} over { sqrt {2} } } f rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } = sqrt {2} f rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } } {} (4.49)

Ea=1π0πωmepsin(ωmet)d(ωmet)=2πωmepEa=1π0πωmepsin(ωmet)d(ωmet)=2πωmep size 12{E rSub { size 8{a} } = { {1} over {π} } Int rSub { size 8{0} } rSup { size 8{π} } {ω rSub { size 8{ ital "me"} } NΦ rSub { size 8{p} } } "sin" $$ω rSub { size 8{ ital "me"} } t$$ d $$ω rSub { size 8{ ital "me"} } t$$ = { {2} over {π} } ω rSub { size 8{ ital "me"} } NΦ rSub { size 8{p} } } {} (4.50)

Ea=(polesπ)pωm=polesNΦp(n30)Ea=(polesπ)pωm=polesNΦp(n30) size 12{E rSub { size 8{a} } = $${ { ital "poles"} over {π} }$$ NΦ rSub { size 8{p} } ω rSub { size 8{m} } = ital "polesN"Φ rSub { size 8{p} } $${ {n} over {"30"} }$$ } {} (4.51)

Ea=(poles)(Cam)Φpωm=(poles60)(Cam)ΦpnEa=(poles)(Cam)Φpωm=(poles60)(Cam)Φpn size 12{E rSub { size 8{a} } = $${ { ital "poles"} over {2π} }$$ $${ {C rSub { size 8{a} } } over {m} }$$ Φ rSub { size 8{p} } ω rSub { size 8{m} } = $${ { ital "poles"} over {"60"} }$$ $${ {C rSub { size 8{a} } } over {m} }$$ Φ rSub { size 8{p} } n} {} (4.52)

§4.7 Torque in Nonsalient-pole Machines

• Consider the elementary smooth-air-gap machine of Fig.4.24 with one winding on the stator and one on the rotor and with θmθm size 12{θ rSub { size 8{m} } } {} being the mechanical angle between the axes of the two windings. These windings are distributed over a number of slots so that their mmf waves can be approximated by space sinusoids. In Fig.4.24a the coil sides s, -s and r, -r mark the positions of the centers of the belts of conductors comprising the distributed windings. An alternative way of drawing these windings is shown in Fig.4.24b, which also shows reference directions for voltages and currents. Here it is assumed that current in the arrow direction produces a magnetic field in the air gap in the arrow direction, so that a single arrow defines reference directions for both current and flux.

Figure 4.24 Elementary two-pole machine with smooth air gap: (a) winding distribution and (b) schematic representation.

• The stator and rotor are concentric cylinders, and slot openings are neglected. Consequently, our elementary model does not include the effects of salient poles, which are investigated in later chapters. We also assume that the reluctances of the stator and rotor iron are negligible. Finally, although Fig.4.34 shows a two-pole machine, we will write the derivations that follow for the general case of a multipole machine, replacing θmθm size 12{θ rSub { size 8{m} } } {} by the electrical rotor angle.

θme=poles2θmθme=poles2θm size 12{θ rSub { size 8{ ital "me"} } = left [ { { ital "poles"} over {2} } right ]θ rSub { size 8{m} } } {} (4.53)

• Based upon these assumptions, the stator and rotor self-inductances LssLss size 12{L rSub { size 8{ ital "ss"} } } {} and LrrLrr size 12{L rSub { size 8{ ital "rr"} } } {} can be seen to be constant, but the stator-to-rotor mutual inductance depends on the electrical angle θmeθme size 12{θ rSub { size 8{ ital "me"} } } {} between the magnetic axes of the stator and rotor windings. The mutual inductance is at its positive maximum when θmeθme size 12{θ rSub { size 8{ ital "me"} } } {}=0 or 2 ππ size 12{π} {}, is zero when θme=±π/2θme=±π/2 size 12{θ rSub { size 8{ ital "me"} } = +- π/2} {}, and is at its negative maximum when θme=±πθme=±π size 12{θ rSub { size 8{ ital "me"} } = +- π} {}. On the assumption of sinusoidal mmf waves and a uniform air gap, the space distribution of the air-gap flux wave is sinusoidal, and the mutual inductance will be of the form

Lsr(θme)=Lsrcos(θme)Lsr(θme)=Lsrcos(θme) size 12{L rSub { size 8{ ital "sr"} } $$θ rSub { size 8{ ital "me"} }$$ =L rSub { size 8{ ital "sr"} } "cos" $$θ rSub { size 8{ ital "me"} }$$ } {} (4.54)

where the script letter E denotes an inductance which is a function of the electrical angle θmeθme size 12{θ rSub { size 8{ ital "me"} } } {}. The italic capital letter L denotes a constant value. Thus Lsr is the magnitude of the mutual inductance; its value when the magnetic axes of the stator and rotor are aligned ( θmeθme size 12{θ rSub { size 8{ ital "me"} } } {}= 0). In terms of the inductances, the stator and rotor flux linkages) λsλs size 12{λ rSub { size 8{s} } } {}and λrλr size 12{λ rSub { size 8{r} } } {}are

λs=Lss+Lsr(θme)ir=Lssis+Lsrcos(θme)irλs=Lss+Lsr(θme)ir=Lssis+Lsrcos(θme)ir size 12{λ rSub { size 8{s} } =L rSub { size 8{ ital "ss"} } +L rSub { size 8{ ital "sr"} } $$θ rSub { size 8{ ital "me"} }$$ i rSub { size 8{r} } =L rSub { size 8{ ital "ss"} } i rSub { size 8{s} } +L rSub { size 8{ ital "sr"} } "cos" $$θ rSub { size 8{ ital "me"} }$$ i rSub { size 8{r} } } {} (4.55)

λr=Lsr(θme)is+Lrrir=Lsrcos(θme)is+Lrrirλr=Lsr(θme)is+Lrrir=Lsrcos(θme)is+Lrrir size 12{λ rSub { size 8{r} } =L rSub { size 8{ ital "sr"} } $$θ rSub { size 8{ ital "me"} }$$ i rSub { size 8{s} } +L rSub { size 8{ ital "rr"} } i rSub { size 8{r} } =L rSub { size 8{ ital "sr"} } "cos" $$θ rSub { size 8{ ital "me"} }$$ i rSub { size 8{s} } +L rSub { size 8{ ital "rr"} } i rSub { size 8{r} } } {} (4.56)

In matrix notation

λsλr=LssLsr(θme)Lsr(θme)Lrrisirλsλr=LssLsr(θme)Lsr(θme)Lrrisir size 12{ left [ matrix { λ rSub { size 8{s} } {} ## λ rSub { size 8{r} } } right ]= left [ matrix { L rSub { size 8{ ital "ss"} } {} # L rSub { size 8{ ital "sr"} } $$θ rSub { size 8{ ital "me"} }$$ {} ## L rSub { size 8{ ital "sr"} } $$θ rSub { size 8{ ital "me"} }$$ {} # L rSub { size 8{ ital "rr"} } {} } right ] left [ matrix { i rSub { size 8{s} } {} ## i rSub { size 8{r} } } right ]} {} (4.57)

The terminal voltages vsvs size 12{v rSub { size 8{s} } } {}and vrvr size 12{v rSub { size 8{r} } } {}are

vs=Rsis+sdtvs=Rsis+sdt size 12{v rSub { size 8{s} } =R rSub { size 8{s} } i rSub { size 8{s} } + { {dλ rSub { size 8{s} } } over { ital "dt"} } } {} (4.58)

vr=Rrir+rdtvr=Rrir+rdt size 12{v rSub { size 8{r} } =R rSub { size 8{r} } i rSub { size 8{r} } + { {dλ rSub { size 8{r} } } over { ital "dt"} } } {} (4.59)

where RsRs size 12{R rSub { size 8{s} } } {}and RrRr size 12{R rSub { size 8{r} } } {} are the resistances of the stator and rotor windings respectively.

• When the rotor is revolving, θmeθme size 12{θ rSub { size 8{ ital "me"} } } {} must be treated as a variable. Differentiation of Eqs.4.56 and 4.57 and substitution of the results in Eqs.4.59 and 4.60 then give

vs=Rsis+Lssdisdt+Lsrcos(θme)dirdtLsrirsin(θme)medtvs=Rsis+Lssdisdt+Lsrcos(θme)dirdtLsrirsin(θme)medt size 12{v rSub { size 8{s} } =R rSub { size 8{s} } i rSub { size 8{s} } +L rSub { size 8{ ital "ss"} } { { ital "di" rSub { size 8{s} } } over { ital "dt"} } +L rSub { size 8{ ital "sr"} } "cos" $$θ rSub { size 8{ ital "me"} }$$ { { ital "di" rSub { size 8{r} } } over { ital "dt"} } - L rSub { size 8{ ital "sr"} } i rSub { size 8{r} } "sin" $$θ rSub { size 8{ ital "me"} }$$ { {dθ rSub { size 8{ ital "me"} } } over { ital "dt"} } } {} (4.60)

vr=Rrir+Lrrdirdt+Lsrcos(θme)dirdtLsrissin(θme)medtvr=Rrir+Lrrdirdt+Lsrcos(θme)dirdtLsrissin(θme)medt size 12{v rSub { size 8{r} } =R rSub { size 8{r} } i rSub { size 8{r} } +L rSub { size 8{ ital "rr"} } { { ital "di" rSub { size 8{r} } } over { ital "dt"} } +L rSub { size 8{ ital "sr"} } "cos" $$θ rSub { size 8{ ital "me"} }$$ { { ital "di" rSub { size 8{r} } } over { ital "dt"} } - L rSub { size 8{ ital "sr"} } i rSub { size 8{s} } "sin" $$θ rSub { size 8{ ital "me"} }$$ { {dθ rSub { size 8{ ital "me"} } } over { ital "dt"} } } {} (4.61)

medt=ωme=poles2ωmmedt=ωme=poles2ωm size 12{ { {dθ rSub { size 8{ ital "me"} } } over { ital "dt"} } =ω rSub { size 8{ ital "me"} } = left [ { { ital "poles"} over {2} } right ]ω rSub { size 8{m} } } {} (4.62)

is the instantaneous speed in electrical radians per second. In a two-pole machine, θme and ωme are equal to the instantaneous shaft angle θm and the shaft speed ωm respectively. In a multipole machine, they are related by Eqs.4.54 and 4.46. The second and third terms on the fight-hand sides of Eqs.4.61 and 4.62 are L(di/dt) induced voltages like those induced in stationary coupled circuits such as the windings of transformers. The fourth terms are caused by mechanical motion and are proportional to the instantaneous speed. These are the speed voltage terms which correspond to the interchange of power between the electric and mechanical systems.

• The electromechanical torque can be found from the coenergy.

Wfld'=12Lssis2+12Lrrir2+Lsrisircosθme=12Lssis2+12Lrrir2+Lsrisircos(poles2)θmWfld'=12Lssis2+12Lrrir2+Lsrisircosθme=12Lssis2+12Lrrir2+Lsrisircos(poles2)θmalignl { stack { size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } = { {1} over {2} } L rSub { size 8{ ital "ss"} } i rSub { size 8{s} } rSup { size 8{2} } + { {1} over {2} } L rSub { size 8{ ital "rr"} } i rSub { size 8{r} } rSup { size 8{2} } +L rSub { size 8{ ital "sr"} } i rSub { size 8{s} } i rSub { size 8{r} } "cos"θ rSub { size 8{ ital "me"} } } {} # " "= { {1} over {2} } L rSub { size 8{ ital "ss"} } i rSub { size 8{s} } rSup { size 8{2} } + { {1} over {2} } L rSub { size 8{ ital "rr"} } i rSub { size 8{r} } rSup { size 8{2} } +L rSub { size 8{ ital "sr"} } i rSub { size 8{s} } i rSub { size 8{r} } "cos" left [ $${ { ital "poles"} over {2} }$$ θ rSub { size 8{m} } right ] {} } } {} (4.63)

Note that the coenergy of Eq.4.63 has been expressed specifically in terms of the shaft angle θmθm size 12{θ rSub { size 8{m} } } {} because the torque expression requires that the torque be obtained from the derivative of the coenergy with respect to the spatial angle θmθm size 12{θ rSub { size 8{m} } } {} and not with respect to the electrical angle θmeθme size 12{θ rSub { size 8{ ital "me"} } } {}. Thus,

T=Wfld'(is,ir,θm)θmis,ir=poles2Lsrisirsinpoles2θm=poles2LsrisirsinθmeT=Wfld'(is,ir,θm)θmis,ir=poles2Lsrisirsinpoles2θm=poles2Lsrisirsinθmealignl { stack { size 12{T= { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } $$i rSub { size 8{s} } ,i rSub { size 8{r} } ,θ rSub { size 8{m} }$$ } over { partial θ rSub { size 8{m} } } } \rline rSub { size 8{i rSub { size 6{s} } ,i rSub { size 6{r} } } } = - left [ { { ital "poles"} over {2} } right ]L rSub { ital "sr"} size 12{i rSub {s} } size 12{i rSub {r} } size 12{"sin" left [ { { ital "poles"} over {2} } θ rSub {m} right ]}} {} # size 12{" "= - left [ { { ital "poles"} over {2} } right ]L rSub { size 8{ ital "sr"} } i rSub { size 8{s} } i rSub { size 8{r} } "sin"θ rSub { size 8{ ital "me"} } } {} } } {} (4.64)

where T is the electromechanical torque acting to accelerate the rotor (i.e., a positive torque acts to increase θmθm size 12{θ rSub { size 8{m} } } {}). The negative sign in Eq.4.64 means that the electromechanical torque acts in the direction to bring the magnetic fields of the stator and rotor into alignment.

4.7.2 Magnetic Field Viewpoint

• In the discussion of Section 4.7.1 the characteristics of a rotating machine as viewed from its electric and mechanical terminals have been expressed in terms of its winding inductances. This viewpoint gives little insight into the physical phenomena which occur within the machine. In this section, we will explore an alternative formulation in terms of the interacting magnetic fields.

Figure 4.25 Simplified two-pole machine: (a) elementary model and

(b) vector diagram of mmf waves. Torque is produced by the tendency of the rotor and stator magnetic fields to align. Note that these figures are drawn with δsrδsr size 12{δ rSub { size 8{ ital "sr"} } } {} positive, i.e., with the rotor mmf wave leading that of the stator FsFs size 12{F rSub { size 8{s} } } {}.

• As we have seen, currents in the machine windings create magnetic flux in the air gap between the stator and rotor, the flux paths being completed through the stator and rotor iron. This condition corresponds to the appearance of magnetic poles on both the stator and the rotor, centered on their respective magnetic axes, as shown in Fig.4.35a for a two-pole machine with a smooth air gap. Torque is produced by the tendency of the two component magnetic fields to line up their magnetic axes.
• We shall derive an expression for the magnetic coenergy stored in the air gap in terms of the stator and rotor mmfs and the angle δsrδsr size 12{δ rSub { size 8{ ital "sr"} } } {} between their magnetic axes. The torque can then be found from the partial derivative of the coenergy with respect to angle δsrδsr size 12{δ rSub { size 8{ ital "sr"} } } {}.
• The line integral of Hag across the gap then is simply HaggHagg size 12{H rSub { size 8{ ital "ag"} } g} {} and equals the resultant air-gap mmf FsrFsr size 12{F rSub { size 8{ ital "sr"} } } {} produced by the stator and rotor windings; thus

Hagg=FsrHagg=Fsr size 12{H rSub { size 8{ ital "ag"} } g=F rSub { size 8{ ital "sr"} } } {} (4.65)

• The mmf waves of the stator and rotor are spatial sine waves with δsrδsr size 12{δ rSub { size 8{ ital "sr"} } } {} being the phase angle between their magnetic axes in electrical degrees. They can be represented by the space vectors FsFs size 12{F rSub { size 8{s} } } {} and FrFr size 12{F rSub { size 8{r} } } {} drawn along the magnetic axes of the stator- and rotor mmf waves respectively. The resultant mmf FsrFsr size 12{F rSub { size 8{ ital "sr"} } } {} acting across the air gap, also a sine wave, is their vector sum.

Fsr2=Fs2+Fr2+2FsFrcosδsrFsr2=Fs2+Fr2+2FsFrcosδsr size 12{F rSub { size 8{ ital "sr"} } rSup { size 8{2} } =F rSub { size 8{s} } rSup { size 8{2} } +F rSub { size 8{r} } rSup { size 8{2} } +2F rSub { size 8{s} } F rSub { size 8{r} } "cos"δ rSub { size 8{ ital "sr"} } } {} (4.66)

The resultant radial HagHag size 12{H rSub { size 8{ ital "ag"} } } {}field is a sinusoidal space wave whose peak value Hag,peakHag,peak size 12{H rSub { size 8{ ital "ag", ital "peak"} } } {} is, from Eq.4.65,

(Hag)peak=Fsrg(Hag)peak=Fsrg size 12{ $$H rSub { size 8{ ital "ag"} }$$ rSub { size 8{ ital "peak"} } = { {F rSub { size 8{ ital "sr"} } } over {g} } } {} (4.67)

Average coenergy density = μo2(Hag)peak22=μo4Fsrg2μo2(Hag)peak22=μo4Fsrg2 size 12{ { {μ rSub { size 8{o} } } over {2} } left [ { { $$H rSub { size 8{ ital "ag"} }$$ rSub { size 8{ ital "peak"} } rSup { size 8{2} } } over {2} } right ]= { {μ rSub { size 8{o} } } over {4} } left [ { {F rSub { size 8{ ital "sr"} } } over {g} } right ] rSup { size 8{2} } } {} (4.68)

Wfld'=(average coenergy density)(volume of air gap)=μo4Fsrg2πDlg=μoπDl4gFsr2Wfld'=(average coenergy density)(volume of air gap)=μo4Fsrg2πDlg=μoπDl4gFsr2alignl { stack { size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } = $$"average coenergy density"$$ $$"volume of air gap"$$ } {} # " "= { {μ rSub { size 8{o} } } over {4} } left [ { {F rSub { size 8{ ital "sr"} } } over {g} } right ] rSup { size 8{2} } πD"lg"= { {μ rSub { size 8{o} } π ital "Dl"} over {4g} } F rSub { size 8{ ital "sr"} } rSup { size 8{2} } {} } } {} (4.69)

Wfld'=μoπDl4g(Fs2+Fr2+2FsFrcosδsr)Wfld'=μoπDl4g(Fs2+Fr2+2FsFrcosδsr) size 12{ { {W}} sup { ' } rSub { size 8{ ital "fld"} } = { {μ rSub { size 8{o} } π ital "Dl"} over {4g} } $$F rSub { size 8{s} } rSup { size 8{2} } +F rSub { size 8{r} } rSup { size 8{2} } +2F rSub { size 8{s} } F rSub { size 8{r} } "cos"δ rSub { size 8{ ital "sr"} }$$ } {} (4.70)

For a two-pole machine

T=Wfld'δsrFs,Fr=μoπDl2gFsFrsinδsrT=Wfld'δsrFs,Fr=μoπDl2gFsFrsinδsr size 12{T= { { partial { {W}} sup { ' } rSub { size 8{ ital "fld"} } } over { partial δ rSub { size 8{ ital "sr"} } } } \rline rSub { size 8{F rSub { size 6{s} } ,F rSub { size 6{r} } } } = - left [ { {μ rSub {o} size 12{π ital "Dl"}} over {2g} } right ]F rSub {s} size 12{F rSub {r} } size 12{"sin"δ rSub { ital "sr"} }} {} (4.71)

The general expression for the torque for a multipole machine is

T=poles2μoπDl2gFsFrsinδsrT=poles2μoπDl2gFsFrsinδsr size 12{T= - left [ { { ital "poles"} over {2} } right ] left [ { {μ rSub { size 8{o} } π ital "Dl"} over {2g} } right ]F rSub { size 8{s} } F rSub { size 8{r} } "sin"δ rSub { size 8{ ital "sr"} } } {} (4.72)

• In this equation, δsrδsr size 12{δ rSub { size 8{ ital "sr"} } } {} is the electrical space-phase angle between the rotor and stator mmf waves and the torque T acts in the direction to accelerate the rotor. Thus when δsrδsr size 12{δ rSub { size 8{ ital "sr"} } } {} is positive, the torque is negative and the machine is operating as a generator.
• Similarly, a negative value of δsrδsr size 12{δ rSub { size 8{ ital "sr"} } } {} corresponds to positive torque and, correspondingly, motor action. The torque, acting to accelerate the rotor, can then be expressed in terms of the resultant mmf wave FsrFsr size 12{F rSub { size 8{ ital "sr"} } } {}; thus

T=poles2μoπDl2gFsFsrsinδsT=poles2μoπDl2gFsFsrsinδs size 12{T= - left [ { { ital "poles"} over {2} } right ] left [ { {μ rSub { size 8{o} } π ital "Dl"} over {2g} } right ]F rSub { size 8{s} } F rSub { size 8{ ital "sr"} } "sin"δ rSub { size 8{s} } } {} (4.73)

T=poles2μoπDl2gFrFsrsinδrT=poles2μoπDl2gFrFsrsinδr size 12{T= - left [ { { ital "poles"} over {2} } right ] left [ { {μ rSub { size 8{o} } π ital "Dl"} over {2g} } right ]F rSub { size 8{r} } F rSub { size 8{ ital "sr"} } "sin"δ rSub { size 8{r} } } {} (4.74)

T=poles2πDl2BsrFrsinδrT=poles2πDl2BsrFrsinδr size 12{T= left [ { { ital "poles"} over {2} } right ] left [ { {π ital "Dl"} over {2} } right ]B rSub { size 8{ ital "sr"} } F rSub { size 8{r} } "sin"δ rSub { size 8{r} } } {} (4.75)

• One of the inherent limitations in the design of electromagnetic apparatus is the saturation flux density of magnetic materials. Because of saturation in the armature teeth the peak value BsrBsr size 12{B rSub { size 8{ ital "sr"} } } {} of the resultant flux-density wave in the air gap is limited to about 1.5 to 2.0T. The maximum permissible value of the winding current, and hence the corresponding mmf wave, is limited by the temperature rise of the winding and other design requirements. Because the resultant flux density and mmf appear explicitly in Eq.4.75, this equation is in a convenient form for design purposes and can be used to estimate the maximum torque which can be obtained from a machineof a given size.

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